Bailey pairs

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31: Bibliography

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A trinomial analogue of Bailey’s lemma and $N=2$ superconformal invariance. Comm. Math. Phys. 192 (2), pp. 245–260.

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External Links:

ISSN 0010-3616, Document, MathReview (Anne Schilling), ZentralBlatt

Referenced by:

§17.12.

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BibTeX

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Umbral calculus, Bailey chains, and pentagonal number theorems. J. Combin. Theory Ser. A 91 (1-2), pp. 464–475.

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Notes:

In memory of Gian-Carlo Rota

External Links:

ISSN 0097-3165, Document, MathReview (Alessandro Di Bucchianico), ZentralBlatt

Referenced by:

§17.12.

Encodings:

BibTeX

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Bailey’s Transform, Lemma, Chains and Tree. In Special Functions 2000: Current Perspective and Future Directions (Tempe, AZ), J. Bustoz, M. E. H. Ismail, and S. K. Suslov (Eds.), NATO Sci. Ser. II Math. Phys. Chem., Vol. 30, pp. 1–22.

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External Links:

ISBN 0-7923-7119-4, MathReview, ZentralBlatt

Referenced by:

§17.12.

Encodings:

BibTeX

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32: 1.13 Differential Equations

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Fundamental Pair

►Two solutions $w_1(z)$ and $w_2(z)$ are called a fundamental pair if any other solution $w(z)$ is expressible as …A fundamental pair can be obtained, for example, by taking any $z_0\in D$ and requiring that … ►The following three statements are equivalent: $w_1(z)$ and $w_2(z)$ comprise a fundamental pair in $D$; $𝒲\left\{w_1(z),w_2(z)\right\}$ does not vanish in $D$; $w_1(z)$ and $w_2(z)$ are linearly independent, that is, the only constants $A$ and $B$ such that … ►If $w_0(z)$ is any one solution, and $w_1(z)$, $w_2(z)$ are a fundamental pair of solutions of the corresponding homogeneous equation (1.13.1), then every solution of (1.13.8) can be expressed as …

33: 15.11 Riemann’s Differential Equation

… ►Here $\{a_1,a_2\}$, $\{b_1,b_2\}$, $\{c_1,c_2\}$ are the exponent pairs at the points $\alpha$, $\beta$, $\gamma$, respectively. Cases in which there are fewer than three singularities are included automatically by allowing the choice $\{0,1\}$ for exponent pairs. …

34: 26.15 Permutations: Matrix Notation

… ►The inversion number of $\sigma$ is a sum of products of pairs of entries in the matrix representation of $\sigma$: … ► $N_k(B)$ is the number of permutations in ${𝔖}_n$ for which exactly $k$ of the pairs $(j,\sigma (j))$ are elements of $B$. …

35: 36.7 Zeros

… ►Inside the cusp, that is, for , the zeros form pairs lying in curved rows. … ►Away from the $z$-axis and approaching the cusp lines (ribs) (36.4.11), the lattice becomes distorted and the rings are deformed, eventually joining to form “hairpins” whose arms become the pairs of zeros (36.7.1) of the cusp canonical integral. …

36: 8.13 Zeros

… ►When $x>x_n^{\ast }$ a pair of conjugate trajectories emanate from the point $a=a_n^{\ast }$ in the complex $a$-plane. …

37: 15.19 Methods of Computation

… ►This is because the linear transformations map the pair $\{{\mathrm{e}}^{\pi \mathrm{i}/3},{\mathrm{e}}^{-\pi \mathrm{i}/3}\}$ onto itself. …

38: 23.20 Mathematical Applications

… ►The two pairs of edges $[0,{\omega }_1]\cup [{\omega }_1,2{\omega }_3]$ and $[2{\omega }_3,2{\omega }_3-{\omega }_1]\cup [2{\omega }_3-{\omega }_1,0]$ of $R$ are each mapped strictly monotonically by $\mathrm{\wp }$ onto the real line, with $0\to \mathrm{\infty }$, ${\omega }_1\to e_1$, $2{\omega }_3\to -\mathrm{\infty }$; similarly for the other pair of edges. For each pair of edges there is a unique point $z_0$ such that $\mathrm{\wp }\left(z_0\right)=0$. …

39: 2.7 Differential Equations

… ►One pair of independent solutions of the equation …In theory either pair may be used to construct any other solution …This kind of cancellation cannot take place with $w_1(z)$ and $w_2(z)$, and for this reason, and following Miller (1950), we call $w_1(z)$ and $w_2(z)$ a numerically satisfactory pair of solutions. … ►This is characteristic of numerically satisfactory pairs. … ►In oscillatory intervals, and again following Miller (1950), we call a pair of solutions numerically satisfactory if asymptotically they have the same amplitude and are $\frac{1}{2}\pi$ out of phase.

40: 12.2 Differential Equations

… ►For real values of $z$ $(=x)$, numerically satisfactory pairs of solutions (§2.7(iv)) of (12.2.2) are $U(a,x)$ and $V(a,x)$ when $x$ is positive, or $U(a,-x)$ and $V(a,-x)$ when $x$ is negative. For (12.2.3) $W(a,x)$ and $W(a,-x)$ comprise a numerically satisfactory pair, for all $x\in ℝ$. … ►In $ℂ$, for $j=0,1,2,3$, $U({(-1)}^{j-1}a,{(-\mathrm{i})}^{j-1}z)$ and $U({(-1)}^{j}a,{(-\mathrm{i})}^{j}z)$ comprise a numerically satisfactory pair of solutions in the half-plane $\frac{1}{4}(2j-3)\pi \le \mathrm{ph}z\le \frac{1}{4}(2j+1)\pi$. …